{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "G6szyyVwx7UM"
      },
      "source": [
        "# What is Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5bXXHfNjx7UP"
      },
      "source": [
        "Linear Regression is a statistical and machine learning technique for predicting a continuous outcome variable (also called the dependent variable) based on one or more predictor variables (also called independent variables).\n",
        "\n",
        "The goal of linear regression is to find the best-fitting straight line through the data points. The best-fitting line is called the regression line.\n",
        "\n",
        "The equation of a linear regression model is:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "s81qehiCx7UQ",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "y = β0 + β1*x1 + β2*x2 + ... + βn*xn + ε"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XNlEUVZmx7US"
      },
      "source": [
        "\n",
        "\n",
        "Here:\n",
        "\n",
        "- `y` is the outcome variable we want to predict.\n",
        "- `x1, x2, ..., xn` are the predictor variables.\n",
        "- `β0, β1, ..., βn` are the parameters of the model that we need to estimate. `β0` is the y-intercept and `β1, ..., βn` are the coefficients of the predictor variables.\n",
        "- `ε` is the error term, the part of `y` that cannot be explained by the predictors.\n",
        "\n",
        "The coefficients `β0, β1, ..., βn` are estimated using the method of least squares, which minimizes the sum of the squared residuals (the differences between the observed and predicted values).\n",
        "\n",
        "Linear regression makes several assumptions, including linearity, independence, homoscedasticity, and normality. Violations of these assumptions can lead to inaccurate or misleading results."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2qlDVBtmx7UT"
      },
      "source": [
        "# Simple Linear Regression?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "i3cFuB44x7UT",
        "outputId": "706746bf-6587-40f3-8848-dbc5c80dfd19",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [
        {
          "data": {
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",
            "text/plain": [
              "<IPython.core.display.Image object>"
            ]
          },
          "execution_count": 4,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "from IPython.display import Image\n",
        "\n",
        "Image(filename='./images/LR.jpg')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CF2dVoU8x7UU"
      },
      "source": [
        "Simple Linear Regression is a type of linear regression where we have a single independent variable to predict the dependent variable. It is the simplest form of linear regression that shows the relationship between two variables.\n",
        "\n",
        "The goal of simple linear regression is to find the best-fitting straight line through the data points. The best-fitting line is called the regression line.\n",
        "\n",
        "The equation of a simple linear regression model is:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rn_0aKgtx7UV",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "y = β0 + β1*x + ε"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XUkO7snZx7UV"
      },
      "source": [
        "\n",
        "\n",
        "Here:\n",
        "\n",
        "- `y` is the dependent variable we want to predict.\n",
        "- `x` is the independent variable used to predict `y`.\n",
        "- `β0` and `β1` are the parameters of the model that we need to estimate. `β0` is the y-intercept and `β1` is the slope of the line.\n",
        "- `ε` is the error term, the part of `y` that cannot be explained by `x`.\n",
        "\n",
        "The parameters `β0` and `β1` are estimated using the method of least squares, which minimizes the sum of the squared residuals (the differences between the observed and predicted values).\n",
        "\n",
        "Simple linear regression makes several assumptions, including linearity, independence, homoscedasticity (constant variance of the errors), and normality of the errors. Violations of these assumptions can lead to inaccurate or misleading results."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FND0_0k7x7UV"
      },
      "source": [
        "## Random Error(Residuals)?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AIh_lQ9Ox7UW"
      },
      "source": [
        "In the context of regression analysis, the random error, also known as the residual, is the difference between the observed value and the predicted value of the dependent variable.\n",
        "\n",
        "For each data point, the residual is calculated as:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rDYX9D31x7UW",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "e_i = y_i - ŷ_i"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "r3VoY7J1x7UX"
      },
      "source": [
        "\n",
        "\n",
        "where:\n",
        "- `e_i` is the residual for the i-th data point,\n",
        "- `y_i` is the observed value of the dependent variable for the i-th data point,\n",
        "- `ŷ_i` is the predicted value of the dependent variable for the i-th data point.\n",
        "\n",
        "Residuals are used to assess the goodness-of-fit of a regression model. If the model fits the data well, the residuals should be randomly scattered around zero, and their distribution should be approximately normal.\n",
        "\n",
        "Residuals are considered as a source of randomness in the model, hence the term \"random error\". They capture the variability in the dependent variable that cannot be explained by the independent variables in the model."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ICKZCvfGx7UX"
      },
      "source": [
        "# What is the best fit line?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BIJtARe1x7UX"
      },
      "source": [
        "The \"best fit line\", also known as the line of best fit or the regression line, is a line that is used in regression analysis to approximate the relationship between the independent and dependent variables in a dataset.\n",
        "\n",
        "In the context of linear regression, the best fit line is the line that minimizes the sum of the squared differences (also known as residuals or errors) between the observed values (actual values) and the predicted values (values predicted by the line) of the dependent variable.\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Sc8pZPF4x7UY"
      },
      "source": [
        "# Cost Function for Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DKuuYhOfx7UY"
      },
      "source": [
        "The cost function for linear regression, often referred to as the Mean Squared Error (MSE), measures the average squared difference between the actual and predicted values. It is used to find the best possible linear line that fits the data.\n",
        "\n",
        "Given a dataset with `n` points, `(x1, y1), (x2, y2), ..., (xn, yn)`, and a linear model `y = β0 + β1*x`, the cost function `J(β0, β1)` is defined as:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "qrbp-wflx7UY",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "J(β0, β1) = 1/(2n) * Σ[i=1 to n] (yi - (β0 + β1*xi))^2"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wF69lgvNx7UZ"
      },
      "source": [
        "\n",
        "\n",
        "Here:\n",
        "\n",
        "- `yi` is the actual value of the dependent variable for the i-th data point.\n",
        "- `β0 + β1*xi` is the predicted value of the dependent variable for the i-th data point.\n",
        "- The term `(yi - (β0 + β1*xi))^2` is the squared difference between the actual and predicted values for the i-th data point.\n",
        "- The sum `Σ[i=1 to n]` adds up these squared differences for all `n` data points.\n",
        "- The term `1/(2n)` takes the average of these squared differences and divides by 2 (the 2 is used for mathematical convenience when computing the gradient).\n",
        "\n",
        "The goal of linear regression is to find the parameters `β0` and `β1` that minimize this cost function. This is typically done using a technique called gradient descent."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bdbMF5YCx7UZ"
      },
      "source": [
        "# Gradient Descent for Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "j2zzN_bpx7UZ"
      },
      "source": [
        "Gradient Descent is an optimization algorithm used to minimize the cost function in linear regression (and many other machine learning algorithms). The goal is to find the model parameters (coefficients) that minimize the cost function.\n",
        "\n",
        "Here's a simplified explanation of how gradient descent works in the context of linear regression:\n",
        "\n",
        "1. Initialize the model parameters with some values. These could be random values or all zeros, for example.\n",
        "\n",
        "2. Calculate the cost function with the current parameters.\n",
        "\n",
        "3. Compute the gradient of the cost function at the current parameters. The gradient is a vector that points in the direction of the greatest increase of the function. It is calculated by taking the partial derivative of the cost function with respect to each parameter.\n",
        "\n",
        "4. Update the parameters by taking a step in the direction opposite to the gradient. This is done by subtracting the gradient times a learning rate from the current parameters.\n",
        "\n",
        "5. Repeat steps 2-4 until the cost function converges to the minimum.\n",
        "\n",
        "The learning rate is a hyperparameter that determines the size of the step. If it's too small, the algorithm will take a long time to converge. If it's too large, the algorithm might overshoot the minimum and never converge.\n",
        "\n",
        "In the case of linear regression with a cost function `J(β0, β1)`, the update rules for the parameters are:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "IUWZGZ8wx7UZ",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "β0 = β0 - α * (1/n) * Σ[i=1 to n] (ŷi - yi)\n",
        "β1 = β1 - α * (1/n) * Σ[i=1 to n] ((ŷi - yi) * xi)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Qxo1bQDfx7Ua"
      },
      "source": [
        "\n",
        "\n",
        "where:\n",
        "- `β0` and `β1` are the parameters,\n",
        "- `α` is the learning rate,\n",
        "- `n` is the number of data points,\n",
        "- `ŷi` is the predicted value for the i-th data point (`β0 + β1*xi`),\n",
        "- `yi` is the actual value for the i-th data point,\n",
        "- `xi` is the value of the independent variable for the i-th data point."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tklhNxilx7Ua"
      },
      "source": [
        "## Gradient Descent Example\n",
        "Let’s take an example to understand this. Imagine a U-shaped pit. And you are standing at the uppermost point in the pit, and your motive is to reach the bottom of the pit. Suppose there is a treasure at the bottom of the pit, and you can only take a discrete number of steps to reach the bottom. If you opted to take one step at a time, you would get to the bottom of the pit in the end but, this would take a longer time. If you decide to take larger steps each time, you may achieve the bottom sooner but, there’s a probability that you could overshoot the bottom of the pit and not even near the bottom. In the gradient descent algorithm, the number of steps you’re taking can be considered as the learning rate, and this decides how fast the algorithm converges to the minima."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "yFASR-2bx7Ua",
        "outputId": "20e91b4b-26cc-4546-a4ce-2b2cb142679f",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [
        {
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xWc9Edk0M+xWrfBnmZ7v3Ajt2ld00fiSQklwPHZt4eCPvXeRScSY1/jbOkxTJIwOyTE9/CI9i22WnrP6YQfAgDdfM7dpXdNH4GK/xNeYtT4ycnhEIKVcihYSyuP8ATOa+TBhsyEaocmk5MACNjbZnlt+F5iRaa8hLR9wQbeiEIxMAZH+mHKj2AkT0Zedg3R9eYKGcD4qEC4nxvzJYi0aDnj1oRJLng3u8D7FuZ8sPOOGmyq0TTc93HGrpvP8Ad629AxOq/f8Ak1tsGYnDOsR6DevPv6KH8jrFg61L+Z1i69fM7TpXadPyCWwWsyLYVczDQyWIii5EzPzpcviurPMeYtaEi5Bp0hAoQHHFxFXT0CDqd6+g77d+cmJbBaE5uBhe/wA+icDed2fQ+a7TpXadPBZZ498c8OlGvYbDiM6Uphh9j6pIkATH6KXOKLwOAyPJ97CuhWgmy9ivgph/NBwwXTMt2sWXUCf5COVCSHAEB6Dvt3gKG8mwLWJ5tHFJZcjGfbn6Cz+RrFg61OzqAuvXze06V2nTwYOPDWLTcbTmL0DAKxGrigWshvXBFJokHRJ4dzFKK9xUB1bVwvr+q/jZdC1bYACPyeW/wXXkS8qHGIj0PfbvApImIkjUvGWuLFbRIeK1oIIax55wMbzuz6HzjlafOowdPi8Ok7Hrq+XCaueTny3LHlWCFhNRzIOiSss2LDifdCiGOBe6FZ8DV+GNZ8DtHGoYDxjq+O/s/tM23Lf0XWS93wprHUKkItkmGdqwWLB82EHUBYKlp6rLr187umqu3aeFJIblTCYyTq4aLOLQxA6PmcK2i3v+q9hUB0Ki7Czh7VGnQY4cDn5F2pQTuPtKgvOAyPRd3v4ruAITFrShE5fMYbb+bd5t90eRbm+f3TVXbtPFB2JMAwN6EhepzORfesFmONz+W5Eya1fXNvBzqSSRiZnHwgyACVcqYmlgPI64/wCej7vfxAwCrI1I4AhYQsTOJRQpGDJz8uPGbguvAL1PRDF8Xfz+6aq7dp4xaG8sDI+r/VRoJAcRzr68P3v71kg1vdH+akcED7fiEVHDG/usGQO46Vjh+bY5HCy5/jj6Ts9/GnqhokLyqHotS2yU5R5SgKsBQcBkblr+XCPQKEO5UIM0/HjBAFWRpiVVd8zXufHD8gEATRqPNboOmFRhE2S6n8VCzckh1KZxi7RGJ7GKgr275hV1Sm5MX1jAqGQXty57cKEAqRwTyXpu7tlz57cfQ961V7T8eQvkMwxWSb1DEYkYaDu3inZy2nB60ZzSKVleJ/ysCHlTDqaXj1zNeKNR9dOmnkIYGy/l2KgdWZeOc+h71qr2n48koyTAxX8bUiCNLkHuHx4ShGlWBQiBcL0nwdiQBcQvXGH0JCNNxVHaHy8R8FWaOhUe2KjA6Nvn0XftVe0/HlDwokuhdRovyPaMOJbhW6r5vw8Z43zac6nRSl07zq9P1je3lG1JCs6OEcL/AEG+PzRFQwST8yBdK3WSoh2IkwdB3Po+x669ueZ340XKlwPQ7otFBXJg9R3bR5it8QvHEweZX2KH2FdGRi8y/vWdkrZ+k7Hrr25+m7to/R9j117c/Td20esNrQDETp44Pr0aqCg+jJIn82rXS53gC5cu/vPytWr+wpd4fdd4fdd4fdaPM8D73v8AGU/xlP8AGU0ud4JVVX/d/ksWHrYCiswYZ96O5Z43Fk69PJNFLZgGaH4/MaN/X+BLFizW5S8BVVX/ABlP8ZT/ABlNTkD+fB73u4fqu8Pqu8PqtflfxeDVq1anK/NXLvfMyEkFXQksb8Bb/H6OCRi5/wB6QxrW7AnP8KnqBxexn+IxvBi9DXxQKfgWXCTL9923SnvZvNJiuwJvSjJED3Ake9qkuoy5JsEp/XeKywXwtE0EFa7MjW+Z+KMtX4PmIZ2iimxblC0atxqfCqlixF9taHxItSOe5feo0K5TdCM4oTIBx1KF0dIoAIyvuYyCza9cCRCExD7qcgAEHVnV81rDhtji2tUbI6OZ0ADrFRyNhjUSOJcjnSasG7dDOmYbcwBp6KPVS8IGV/CEiwIEp0v4R5QgJYCbVHQMCPEYXEDV6Ono1jwMncpzqYTFnGU50JIVfAlwtvWFSQfYRBRJbPaniliS0gw4VRLIiwSMbbUPixNlJVrbVI5GV50jjXMsUmaFCMCyjm0RYvvXXh2/pPtUq9CrCC+D+KvhqEsq3J8TUnTyWYK2tvT3CAAdsExPPWouKkZOzkpagKGETjapQbjkWbosb0W0EMjHC9IUkhF6MypXOIyQEb53dfRd80KQmOIUTfXUYu1XixGDhHOnN1K2hkN496LkAJahbe0UXUi5kw3JoZOO8eMwwN6a6+gnR1ONS/mVErZTBxWoRtIlK3yXIcqWEuQsp9KxTuASEmck50x5WL4Gel2sabdGaEFuNKmNXCMZNJxjFZQHsJ3oaMeNuRs4Vln+Ui26y6n0acAE4jOmmJDmTD0gcMw6wJQAAIDLygQDgoOtExBYDA9OE8UirwKfEAlYimBVoHcvMy5zSYazhWZljfTKhpgBaBVmBipZtbehzyXrmEUlYvi0EirJzTa0aNsooLYAglOtqU0ytA2G+c0BplS6ZyfEVG1QLstrQoilpmI2p3Q9McOlAmBLkGRnWmbNspxEYYG0VbQ2XEiczGk1mcflKAMAIP8Ax3//2gAMAwEAAgADAAAAEMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMjMzMzMzMzMzMyMzMzMzMzMzMwMzMzMzMzMzMzIzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMwMzEyMTMzMzMzMzMzMzMzMzMzAzMjARMzMzMzMzMzMzMzMzMzMDMzMzMzMzMzMzMzMzMzMzMzMwMzMyEzMzMzMzMzMzMzMzMzMzIzMzMDMzMzMzMzMzMzMzMzMzMjMzIyEzMzMzMzMzMzMzMzMzMyMzMzMDMzMzMzMzMzMzMzMzMzMzMyMyEzMzMzMzMzMzMzMzMzMzczMzMjMXMzMzMzMzMzMzszMzKfMTMyEpszMzMw8zMzMzMzMzAyMjMzMDMzMzMzFzMzMzJXMzMDBeMzMzzzMzMzMfMzMzMnMzMwMqwzMx+3MzMzMylzMzMxczMzAz1LMGPp8zMzMzMHMzMzFzMzMzMANDMzMzMzMzMyfzMzJXMzMzMwJDMzMzMzMzMzL/MzMzMzMzMzNHE/N7MzMzMzM3szMwczMzMzMwxY8zMzMzMzMzAzMyGzMzMyMzMzxJczMzMzMzM1MzNzMzMzEzMpswWzMzMzMzMzMzMzMzMzMTMzNzE/MzMzMzMzMzM1szMzMzMzMmv7MzMzMzMzMyqy8zMzMzAzMzJzMzMzMzMzMzMqfzMzMzMDMzMzMzMzMzMzMzMzMzMzMzMwMzMzMzMzMzMzMzMzMzMzMzMzAzMzMzMzMzEzMjMzMzMzMzMzMzMzMzMzMzIzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMRMzMzMzMzMzMzMzMzMzMzMzAgASESITMzMzMTIzAzMzMzMzMyEDIBIjMzMzMzAgASAjMzMzMzMTMzAzMzMzMzABMSEDEzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzMzM/8QAKxEBAAAEBAUEAwADAAAAAAAAAQARITEwQXGBUWGRobEQINHwQMHhgJDx/9oACAEDAQE/EP8AQSC2w1dXPOf4agFZ94oss6NHtCreGWCZLLuhV7Q7XnByifabWwtiO/8AUtsASWqC62Z6ou+DsSRq4QgZpiy0zrrlo4PPxCOfiuBKHRff/kCJM95ORDktWWln7xwk70q7C+MHejv/AFPbBrcAm5U6oG8NU8nt7malfiCFc4lqWBvKveeEvvy80ezE44D7nB+/DBQbwziwxlo1O0DNNbL1BSux8XABQgbAz0KvaFW8s8K8LPCFDJKPWU98DYh3/qW2F1ctrdkNoASYBRf5jbO/8gRI9dkSNX+MQvUysdYGZM93PxCOfiuEgsijuU8LGl8YlvOvGJT2buO1jpLqxAKlMMKtS/upcIerTwrth56ISZ0X+5+gRzPMIEz0BWRCJxajpSe998QUZkTmmXb5myy93NCnsUO+IcJlNqS/XVOJWanf7r1gbIa+DP5+fTYc9y3hjO51BlLWfx7QnQhSlWo6X6s3fE5is1L+Tseg1ZxzI4/JAEWYdx9j9K4xTXYy3y7hPlCSo+wSm5Z7lvDFU7hWBQuVNGp8PP1AwqQZmnKhp9XHojz75966J7KMzc9ih1Z9DGE/mdVfo10Vy9Q0Eat+h0UgAJGPzDc1Lm51QPVwzPs9C7As2KGhQ/vPGA3T7J5NmD4l3cX6cznMMmBVcg4v3kVhE4Ufteb2oZfgGmkkDK6rnY0b8lLj6SqNvpxa5NWSfgU5nzqWd4nz0ABrIAn+EsahojUTgn1LlY82phtI7qcSETzX/J//xAAqEQABAwEGBgMAAwAAAAAAAAABABEhMDFBUXGx8BAgYZHR4UCBoYCQ8f/aAAgBAgEBPxD+hRg/w3u5xJJkC4cUSYmQDBhRI1+lNzifVA0gozIG9u1JqbZRcCmAUAKMoggsechep6BDe8qQOzY8ijJhifdGDHBCDXj95g0Obao5wZGSNIKsCmDQ25Rr/lEhwxRCAb0DWW3+eJAewYrO8fGCJJko2IhAMGFKxEg4rZ3+qE3PX1SgQ+0QnCJyGNF95/PaKbnjMgbj21QNE8uiGLHmcCmAUgDY5Wd4J1hGTd6edOQSZw34qEliyAl8QbOabCncAT8o4CIutEQmPG8xqg8YaPrmEmcN+KgIi99/qe3g/m8uyPbQhiu8eOEh1b0rRjPjmvMakSPp398DQuXQzh4KIbt20VuXe860B0b05bHVvSqwYp4TbxMEkFH4v14Do2OQdmsbS468SJAM0SSXNeYd3Fg5TSbawAYqe+16oYuUYSbT8AgEMU0b4fo3bw7Q19a5fACsCKuLe/whjKdgwyYoAAw/k/8A/8QALBABAAEDAgQGAgMBAQEAAAAAAREAITFBUWFxgfAQIDBAkaFQsWDB0fFw4f/aAAgBAQABPxD/AMZOMEIkSImRPwwinPyU8kgcB+DGYdhcTysg1kOlHBMDZ4A8FW+g2/Ctp6WpYIi83Bss6UsYF1AgtocWzs/lqfFbeZGKvnMCfwzlarc1X9/g+5befLKm36/4wYGJQ3QqUuZENO0iCGUtV6O6ckiWSY3WtU3fMThMbVgsTMUJHUGEsZnUCSztSzksh+EiDdNOMDxQoWEayUROWpld4NSxGtNuCOogMuWTcvRoqPigkRMn5rT3Lbz5YgC834UQItlBvW3Pk3QPsxMpMRQN9Qu2SQAjlATbBJKRhSBbADQLNSHRTgOpBELYpmCX+BGBSmQsJcYilFH3LoETmFWsMyxOTnKFABlKZFBmWBCmywtgdkAFFg3gxeAhC8ABxhEzFo/Nae5bfwrLpZpCI9VEcgAPTgh1GK+kpYBNkaOGgAd8MzDVCmYgUIlg8hcRMn5xd23d6h7eh7AD1IncNPB4JjZVZQ+9XMGSk5UAyDu0afm13bd3pnjDgRsoAc0daeEA7ivUs6+JrLUytXbcbGGgOWlPh8tcNR/NLu27vTPWhYdkb8pzdPLAlYUg8R4GgTxE0sGgINqNnIJBcU9tmaCclT8H33f6VBC0CSVAmA1XAatEDEJsxfv1CHgfMg/CCWU66dMjdDRKIxwBsWE2g5IIn5hJ33f40E2gSo3YzHyefURh0bq5HQjX0J8xYqVmJsrCrDOlTgCu4uVft6hC8gdzJbFyHCAN4nX8MkSAJHqIA9Tvu/xoBMYZMTCEdmeBR0pog7JDDomiYTfyvsG9ABKvCKt3U6IUJtASfY9GL5BDAwa4b9GRSpri6gYxHnDLUeKEnAMzF6DzANaNKEGgIPZJG2YUyUi3FZSQZuM0OFWGTvT2evUOBfnrRzW3qBaCL86S2A3xJ1sIjLr4Eh4vDQJRCRGRPI84Fi1Ah53lbVBBBj0jr7HyhHIjCJcQSh28yyWFaaDDBiHgwJwAUbPMHW39hf8A0vlf78QUUUAboQhBxJioEZskqxwVJw9lYIoLlx1aCrTFLcuerS+p2bfXZtvi1TJtnkgsC14xViuc6tIW2RYL3wzbCzxJ6c/HVwM+joznTamx6lt5pFLEEyXhkpQqDiGGwzAqpF5Lgp4DWRumYgZzq9m+WS9CHiDV+yurSQB5OnA9lZi/JLUfO5jb1ezb67Nt8lpW2hAjowfE3tK0fIz9FzZ2SjLOg2W4aHih2mpiRzw1DaCDu3USZoUAEAcg9FpEWTfvEredouskfmK0MNQ/Nf4JTOkNS6kUVdedqkucGcJENsB596HSsCA4B7R9QbZkRGyk+Og3OrqKQK5eITHsABx4LlwPmgpph53q+UmNiDT1ezb67Nt8gAKmJhNkdEbjolM2YeQMJA2Q4JTSihpAJE4lHQIfzAnMTYtrbhUmMKOrP7w6PlaiGQD5axWOwwui1GZTtP8ASH5rJb1P6N9KOBWBA6HiE4C2ZBV8C+BrUHxBbsBldXd9q+On4GBxGoxPRowQw1AaUALBIAWGJ+T17P6EtR87mNvWsrMmKLozDnp8pSTYA1J60vM3lIpmpBgOIEvg6FBGtSsGV1XK6rUOciOjf6w61jsOLrk8cluBUGFpu43JGvgoI+E/IK+CoDqp+CUZMtd+0fvz6tDsEG1yAO0/ZM3VVzWfKaMEpku5y3Kuoo4gmbGyIjqNGEaZIfVvc0bnfKUl0JdKDKurLvlp5EHr0O+7fKDAIQiSJSVkbn1Fjgk21JqXlYNgdkgXBRqcHEz8JAerWT3gYh6jS07cPyBh4kO80pZ8ufouSzuegx8tkh/SMFAXAuCAgPj2X337eYPDZUlXipYnJIBLEVD0NhqIOKyXCV9W4r7RKR1qPHZ9hQ77t8wCwoJILSBLTnK2oUcNxIhNISp0LtKJnAiZJmxiLfPjlZ8ElZDLwTxMeDfi4p9fEIQuwbjwfK19SEAMq0VAQaE7mNGS+Yezffft5itqgSI5EpmkwAESoJgYmc5UIlYFwhwhuX9OVFDEkQlZsFco1KwhoVKZU6qyrqr7Ch33b54FLlSsWOwRwIdVLHwAFHbIgtWYDonacBWiFtPp/tSyl7DmvCwek0Mklyixqizhs6DgyUIMchM+TL5ycdKEAoRuJ46zgrRy6wlNkOHtO34vOlIADeMIpFnaKCCoGRQkvFg45w+kTIEqsAUZ6Q21xedByDv7BEEIo1EYgDDo85e1QJEciUGc1GUbqbON+JeK1jshI0pldrr/AKV8S7feD8KSQFs/srBzikEJlB4oSZcrcKDTeSzYundydaSJXkxJidgSroC0QsUaQ/GNg0ANPad629BlCYFMBYTcW3cgoygpRIm56KSFsDv+zPZyKAAAALAexPdt2ehkiBuD3VoHuKjJaYIwA9nU1W2XyygeUSU2hD90sFIRCoQWi3us0DQloscyM3AO8m3te9beggiJI6UlDljX18WvT5PO2sWAnXaK68jKUBaTLjLz20ICx7I923Z6LpGZ55PVYVZPmohaizjS/wD0VnSfI5mzsAZVphzMtW++3GW8FFSdIYvED2vetvR0bImSvrMEw2zRfvJ5os2ElOA344CVgKUkkJpHc4bV1X2D2Ns1dZJ2j0uUchKFcfpwyUDhUTEOHdbg6tlCGTSH5PAkhtbJ6oPFBT8AD5OiIQ7pBoTd9tl6DckJ+/ShcSWQVEw2wkuwLhss1G9AmTISdTxXNQIuLjA4jd0FtTKA6ULu6YxK3hfQPZ5e5beo0kuZk5xL81jaWNuSSsIFgl0PwmhZhTPHvefilFgJv+os9KbLV5uYk+Iq501Bycq5V3b+1y9y2/hWjL3Lb+BaA2uWZ4k2l085YIJUDODoSlDiQaeLSSGHiD4tm09gvCH0/Jo0aJLhHAv0/Fu3msA8EfS8jhw4kwDhn6mu9/nXe/zrvf5+R48eS4PLP06/7iv/AHFf+4rLgjjP4/v136MY10BrD0VbgKMpb3h4tLW0PRDxeQ8BAFtoPEkSnsgcE/a8n379nw3hj7Vf9xX/ALiv/wAxeQ8ePJ8Plf2q73+9d7/eu9/vW5Nwh8ThxNYBw79vyN27eS43gz7fjo0NEmhsCy1yOfRqINE+MZPpC7Zz+DWYGQGLk5/npqdsAmJhK0C/g2XRh0SMyiTE5PDhbaaWrKuWN/MZOkIQmEzIxOfzyhTP9mgCw3caYshFoaE3lDLii6tw0FJcpjiUsCQt2Nwqa5ubojNual5ibU+Ux3AozaltbEXo0FstkkkcLirpJjeKaWzKpUoiUBEYnWlCWDsqsiDfqkxU6jIPAQmC4AYES0kHkIAhKwVxBYIUakdSRQEG1IYqC8t6bxRxJIBZDApYFvNWfxOINgZeIuo3jWPtYhlBI1uGYUxFI4WxJxMEyXDAMi9TlWukMICRJBQ5Uy/QuoBgZaXvLSmJO+yQ3xyjEswlm8NA15aYlMsxdxmE9kLN9YemwTkuScfKGfPa7iYlxqHlnIYSVIGrBYqCswgEpcFNNF83Bt5YpGVZs7ezatGsBQJ+aJhyUuFKIEJtvQ6rcgao71syHOhDsoMzzqaBBHmaXXpdjECEhIMQJmsweHiqLIuwtEOwUHEihb8oXzUmDMSKsjlTDBi1pofmHLhCo7BG0XywzNwkZrzolgRJCUG1Nu6tDMzg43cKFwAfjQVyziTooxFftm7gCd1lDoT5YYV625DFKaF2DclxWVJFlWDdfEiCNNEaIl6ZwQoNyKAyRC9KWAK04AmJiES3WLxV9jTC5sWTbYkdGoX7BJLjDcWNYC+Zft6DgBAGKLJhgPZuMYYE3rRJYFJNbxEM/suEoRYgDcZSZptiGE5S4MEmV60ZYiROiKsEVGCWheWOBUEags9KhgfWhsqUUyQBCbFqYlYhAi4XAjPNVru3hBy3CFJZZIqX8AHpKSCXBIiMU0kks0GqRIbJJFARfJgEFQhIuuUl8MTkYnAaAgxtNcpXh+gRXaHYwrszyc4QuJuVBg6ssi5dhJrKlxQRKEhJNjIUnLWpXM2dIQ4GRtemL8griwyCSYzEzmh8OciIqG9lTp7SCHkOIxCGjLCgBAG3pD7uAk4IUoV1gYAYAMHt1pKZWyzcGl7xrTafniWuJmWYiZXKtL7pJWwQDgmUZZo6UMTKkMR2DDahNkjoAD4KOWfLECQQJMrquIilP4QclAGG8K0pX+bZM5plW7E6UBIh6jK3fuepQ9BIiKMVLE2hOtLOHllKWkwnc+KEd/t+Hx5IUQRGgAghECKgmzlmoz7ozjZWHC2pMXpsu6AQhUiTzGj3FwxIACCDYjhFQrjhTaDG0FQyLM2rMEpZFglXOYlmKaHZgLmSTHAThROQYNg/8d//2Q==",
            "text/plain": [
              "<IPython.core.display.Image object>"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Image(filename='./images/GD.jpg')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FxJ5wdHkx7Ua"
      },
      "source": [
        "To update B0 and B1, we take gradients from the cost function. To find these gradients, we take partial derivatives for B0 and B1."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "y0wPB6b4x7Ub",
        "outputId": "b325230a-cef1-4f78-f8f0-4d2148a074da",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [
        {
          "data": {
            "image/jpeg": 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            "text/plain": [
              "<IPython.core.display.Image object>"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Image(filename='./images/GDD.jpg')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "US6_IhFcx7Ub"
      },
      "source": [
        "We need to minimize the cost function J. One of the ways to achieve this is to apply the batch gradient descent algorithm. In batch gradient descent, the values are updated in each iteration. (Last two equations shows the updating of values)\n",
        "\n",
        "The partial derivates are the gradients, and they are used to update the values of B0 and B1. Alpha is the learning rate."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "j9XQ8iEgx7Ub"
      },
      "source": [
        "# Evaluation Metrics for Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "I2iRUHfEx7Uc"
      },
      "source": [
        "There are several evaluation metrics used to assess the performance of a linear regression model:\n",
        "\n",
        "1. **Mean Absolute Error (MAE)**: This is the average of the absolute differences between the predicted and actual values. It gives an idea of how wrong the predictions were.\n",
        "\n",
        "    ```python\n",
        "    from sklearn.metrics import mean_absolute_error\n",
        "    MAE = mean_absolute_error(y_true, y_pred)\n",
        "    ```\n",
        "\n",
        "2. **Mean Squared Error (MSE)**: This is the average of the squared differences between the predicted and actual values. It gives more weight to large differences.\n",
        "\n",
        "    ```python\n",
        "    from sklearn.metrics import mean_squared_error\n",
        "    MSE = mean_squared_error(y_true, y_pred)\n",
        "    ```\n",
        "\n",
        "3. **Root Mean Squared Error (RMSE)**: This is the square root of the MSE. It's in the same units as the output, which can make it easier to interpret than the MSE.\n",
        "\n",
        "    ```python\n",
        "    from math import sqrt\n",
        "    RMSE = sqrt(MSE)\n",
        "    ```\n",
        "\n",
        "4. **R-squared (R²)**: This is the proportion of the variance in the dependent variable that is predictable from the independent variables. It provides a measure of how well the model's predictions fit the data. A value of 1 indicates perfect fit, and a value of 0 indicates no fit.\n",
        "\n",
        "    ```python\n",
        "    from sklearn.metrics import r2_score\n",
        "    r2 = r2_score(y_true, y_pred)\n",
        "    ```\n",
        "\n",
        "5. **Adjusted R-squared**: This is a modified version of R-squared that has been adjusted for the number of predictors in the model. It increases only if the new term improves the model more than would be expected by chance.\n",
        "\n",
        "Each of these metrics has its own strengths and weaknesses, and the choice of which to use depends on the specific problem and goals."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OmygGA1kx7Uc"
      },
      "source": [
        "# Assumptions of Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "61nEhlo1x7Ud"
      },
      "source": [
        "Linear regression makes several key assumptions:\n",
        "\n",
        "1. **Linearity**: The relationship between the independent and dependent variables is linear. This can be checked by plotting the variables against each other and checking for linearity or using statistical tests for linearity.\n",
        "\n",
        "2. **Independence**: The observations are independent of each other. This is important for the training and testing of the model. If observations are not independent, it can lead to issues like overfitting.\n",
        "\n",
        "3. **Homoscedasticity**: The variance of the errors is constant across all levels of the independent variables. This means that the spread of the residuals should be similar across all values of the independent variables. If this is not the case, it can lead to heteroscedasticity.\n",
        "\n",
        "4. **Normality**: The errors (residuals) are normally distributed. This can be checked using a Q-Q plot or statistical tests for normality.\n",
        "\n",
        "5. **No Multicollinearity**: The independent variables should not be too highly correlated with each other. Multicollinearity can lead to unstable estimates of the regression coefficients and make it difficult to determine the effect of individual variables.\n",
        "\n",
        "Violations of these assumptions can lead to biased or inefficient estimates of the regression coefficients and incorrect inferences. Therefore, it's important to check these assumptions when fitting a linear regression model."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "P8aRnoEEx7Ud"
      },
      "source": [
        "# Hypothesis in Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8brWx8gqx7Ud"
      },
      "source": [
        "In the context of linear regression, the hypothesis is a function that we believe is a good predictor for the dependent variable. It's a linear function of the independent variables and the model parameters.\n",
        "\n",
        "For simple linear regression (one independent variable), the hypothesis function `h(x)` is defined as:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "xMKqIcUBx7Ue",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "h(x) = β0 + β1*x"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5KIp5I7Bx7Ue"
      },
      "source": [
        "\n",
        "\n",
        "where:\n",
        "- `x` is the independent variable,\n",
        "- `β0` is the y-intercept of the line (the value of `y` when `x` is 0),\n",
        "- `β1` is the slope of the line (the change in `y` for a one-unit change in `x`).\n",
        "\n",
        "For multiple linear regression (more than one independent variable), the hypothesis function `h(x)` is defined as:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Tv2MxvPsx7Ue",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "h(x) = β0 + β1*x1 + β2*x2 + ... + βn*xn"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2XT18fdUx7Uf"
      },
      "source": [
        "\n",
        "\n",
        "where:\n",
        "- `x1, x2, ..., xn` are the independent variables,\n",
        "- `β0, β1, ..., βn` are the parameters of the model.\n",
        "\n",
        "The goal of linear regression is to find the parameters `β0, β1, ..., βn` that minimize the difference between the predicted values `h(x)` and the actual values `y`. This is typically done by minimizing a cost function, such as the Mean Squared Error (MSE), using an optimization algorithm like gradient descent.\n",
        "\n",
        "You can find more details about the hypothesis function in linear regression in the 01. Linear Regression.ipynb notebook."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rc_XEG0Kx7Uf"
      },
      "source": [
        "# Multiple Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SxOSL8Czx7Uw"
      },
      "source": [
        "Multiple Linear Regression is an extension of Simple Linear Regression that allows for the prediction of a response variable from two or more independent variables.\n",
        "\n",
        "The equation for a multiple linear regression model is:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "-E3nrPW_x7Uw",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "y = β0 + β1*x1 + β2*x2 + ... + βn*xn + ε"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uiBarMQ5x7Ux"
      },
      "source": [
        "\n",
        "\n",
        "Here:\n",
        "\n",
        "- `y` is the dependent variable we want to predict.\n",
        "- `x1, x2, ..., xn` are the independent variables used to predict `y`.\n",
        "- `β0, β1, ..., βn` are the parameters of the model that we need to estimate. `β0` is the y-intercept and `β1, ..., βn` are the coefficients of the independent variables.\n",
        "- `ε` is the error term, the part of `y` that cannot be explained by the independent variables.\n",
        "\n",
        "The assumptions for multiple linear regression are similar to those for simple linear regression:\n",
        "\n",
        "1. **Linearity**: The relationship between each independent variable and the dependent variable is linear.\n",
        "\n",
        "2. **Independence**: The residuals (errors) are independent. This is often checked using the Durbin-Watson test.\n",
        "\n",
        "3. **Homoscedasticity**: The variance of the residuals is constant across all levels of the independent variables.\n",
        "\n",
        "4. **Normality**: The residuals are normally distributed.\n",
        "\n",
        "5. **No Multicollinearity**: The independent variables are not too highly correlated with each other. This can be checked using the Variance Inflation Factor (VIF).\n",
        "\n",
        "Violations of these assumptions can lead to problems like inefficient parameter estimates, incorrect inference, and poor prediction. Therefore, it's important to check these assumptions when fitting a multiple linear regression model."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wJJBDuZ4x7Ux"
      },
      "source": [
        "# Multicollinearity?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "f_N24XUzx7Ux"
      },
      "source": [
        "Multicollinearity is a situation in which two or more independent variables in a multiple regression model are highly correlated with each other. This can make it difficult to determine the effect of each independent variable on the dependent variable and can lead to unstable estimates of the regression coefficients.\n",
        "\n",
        "There are several ways to detect multicollinearity:\n",
        "\n",
        "1. **Correlation Matrix**: You can create a correlation matrix of the independent variables. If the correlation coefficient between any two variables is very high (close to +1 or -1), it indicates a high correlation.\n",
        "\n",
        "2. **Variance Inflation Factor (VIF)**: VIF is a measure of how much the variance of the estimated regression coefficient is increased due to multicollinearity. VIF of 1 indicates no multicollinearity. As a rule of thumb, a VIF value that exceeds 5 or 10 indicates a problematic amount of multicollinearity.\n",
        "\n",
        "Here's how you can calculate VIF in Python:\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "HC6Alpr1x7Ux",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [],
      "source": [
        "from statsmodels.stats.outliers_influence import variance_inflation_factor\n",
        "from statsmodels.tools.tools import add_constant\n",
        "\n",
        "# Add a constant to the DataFrame, because the VIF method expects it\n",
        "df = add_constant(df)\n",
        "\n",
        "vif = pd.DataFrame()\n",
        "vif[\"variables\"] = df.columns\n",
        "vif[\"VIF\"] = [variance_inflation_factor(df.values, i) for i in range(df.shape[1])]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yQOrhg57x7Uy"
      },
      "source": [
        "\n",
        "\n",
        "3. **Condition Index**: The condition index is calculated by taking the square root of the ratio of the largest eigenvalue to each individual eigenvalue – a condition index exceeding 30 indicates multicollinearity.\n",
        "\n",
        "If multicollinearity is found in the data, options to solve it include: removing some of the highly correlated independent variables, linearly combining the independent variables, or using regularization methods."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DFpaWprdx7Uy"
      },
      "source": [
        "# Overfitting and Underfitting in Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kvo-3BIMx7Uy"
      },
      "source": [
        "Overfitting and underfitting are common problems in machine learning, including linear regression.\n",
        "\n",
        "**Overfitting** occurs when the model is too complex and fits the noise in the data rather than the underlying trend. This often happens when the model has too many parameters relative to the number of observations. An overfitted model will have very low error on the training data but high error on the test data or new data, because it has essentially memorized the training data rather than learning the underlying pattern.\n",
        "\n",
        "**Underfitting** occurs when the model is too simple to capture the underlying trend in the data. This often happens when the model has too few parameters. An underfitted model will have high error on both the training data and the test data, because it's unable to capture the relationship between the independent and dependent variables.\n",
        "\n",
        "To avoid overfitting and underfitting:\n",
        "\n",
        "- Use cross-validation to evaluate the model's performance on unseen data.\n",
        "- Regularize the model to prevent it from becoming too complex. Regularization techniques for linear regression include Ridge Regression and Lasso Regression.\n",
        "- Use feature selection to remove irrelevant features.\n",
        "- Collect more data, if possible.\n",
        "- Try different types of models to see which one performs best on your data."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uPLzvBXgx7Uz"
      },
      "source": [
        "# Example:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ME0yz1JWx7Uz",
        "outputId": "5a549adb-dd7a-471e-9654-c9f2e0b7d08a",
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Intercept:  -37.02327770606391\n",
            "Coefficients:  [ 4.48674910e-01  9.72425752e-03 -1.23323343e-01  7.83144907e-01\n",
            " -2.02962058e-06 -3.52631849e-03 -4.19792487e-01 -4.33708065e-01]\n",
            "Mean Absolute Error (MAE):  0.533200130495698\n",
            "Mean Squared Error (MSE):  0.5558915986952422\n",
            "Root Mean Squared Error (RMSE):  0.7455813830127749\n",
            "R-squared (R2 ):  0.5757877060324524\n"
          ]
        }
      ],
      "source": [
        "from sklearn.datasets import fetch_california_housing\n",
        "from sklearn.model_selection import train_test_split\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score\n",
        "from math import sqrt\n",
        "\n",
        "# Load the California Housing dataset\n",
        "california = fetch_california_housing()\n",
        "X = california.data\n",
        "y = california.target\n",
        "\n",
        "# Split the data into training and testing sets\n",
        "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)\n",
        "\n",
        "# Create a Linear Regression object\n",
        "model = LinearRegression()\n",
        "\n",
        "# Train the model using the training data\n",
        "model.fit(X_train, y_train)\n",
        "\n",
        "# Print the coefficients\n",
        "print(\"Intercept: \", model.intercept_)\n",
        "print(\"Coefficients: \", model.coef_)\n",
        "\n",
        "# Use the model to make predictions on the test data\n",
        "y_pred = model.predict(X_test)\n",
        "\n",
        "# Calculate and print the evaluation metrics\n",
        "print(\"Mean Absolute Error (MAE): \", mean_absolute_error(y_test, y_pred))\n",
        "print(\"Mean Squared Error (MSE): \", mean_squared_error(y_test, y_pred))\n",
        "print(\"Root Mean Squared Error (RMSE): \", sqrt(mean_squared_error(y_test, y_pred)))\n",
        "print(\"R-squared (R2 ): \", r2_score(y_test, y_pred))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here's a detailed explanation of each output:\n",
        "\n",
        "1. **Intercept**: This is the y-intercept of the linear regression line, which is the predicted value when all independent variables are 0. In this case, the intercept is -37.02327770606391.\n",
        "\n",
        "2. **Coefficients**: These are the coefficients of the independent variables in the regression equation. Each coefficient represents the change in the dependent variable for a one-unit change in the corresponding independent variable, assuming all other variables are held constant. In this case, the coefficients are [ 4.48674910e-01, 9.72425752e-03, -1.23323343e-01, 7.83144907e-01, -2.02962058e-06, -3.52631849e-03, -4.19792487e-01, -4.33708065e-01].\n",
        "\n",
        "3. **Mean Absolute Error (MAE)**: This is the average of the absolute differences between the predicted and actual values. It measures the average magnitude of the errors in a set of predictions, without considering their direction. In this case, the MAE is 0.533200130495698.\n",
        "\n",
        "4. **Mean Squared Error (MSE)**: This is the average of the squared differences between the predicted and actual values. It places more weight on large errors because they are squared before they are averaged. In this case, the MSE is 0.5558915986952422.\n",
        "\n",
        "5. **Root Mean Squared Error (RMSE)**: This is the square root of the MSE. It measures the standard deviation of the residuals (prediction errors). In this case, the RMSE is 0.7455813830127749.\n",
        "\n",
        "6. **R-squared (R2)**: This is the proportion of the variance in the dependent variable that is predictable from the independent variables. It ranges from 0 to 1, with 1 indicating that the model perfectly predicts the dependent variable and 0 indicating that the model does not predict the dependent variable at all. In this case, the R2 is 0.5757877060324524, which means that about 57.58% of the variance in the dependent variable can be predicted from the independent variables."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MnIGcUstx7U0"
      },
      "source": [
        "\n",
        "\n",
        "This code does the same thing as the previous example, but it uses the California housing dataset instead of the Boston housing dataset. The `fetch_california_housing` function loads the California housing dataset."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# R Code Implementation:"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here's an example of how you can implement Linear Regression using the mtcars dataset in R. This example also calculates and displays various metrics such as R-squared, Adjusted R-squared, and Residual Standard Error.\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "vscode": {
          "languageId": "r"
        }
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "Loading required package: ggplot2\n",
            "\n",
            "Loading required package: lattice\n",
            "\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "\n",
              "Call:\n",
              "lm(formula = mpg ~ ., data = train)\n",
              "\n",
              "Residuals:\n",
              "    Min      1Q  Median      3Q     Max \n",
              "-3.4206 -1.5572 -0.3732  1.0568  4.6085 \n",
              "\n",
              "Coefficients:\n",
              "              Estimate Std. Error t value Pr(>|t|)  \n",
              "(Intercept)  9.7756892 21.5000974   0.455   0.6551  \n",
              "cyl          0.0002296  1.2150154   0.000   0.9999  \n",
              "disp         0.0152289  0.0211242   0.721   0.4808  \n",
              "hp          -0.0197934  0.0260743  -0.759   0.4582  \n",
              "drat         0.9656872  1.9744279   0.489   0.6310  \n",
              "wt          -3.9435481  2.1083598  -1.870   0.0787 .\n",
              "qsec         0.9155431  0.8525039   1.074   0.2979  \n",
              "vs           0.4816026  2.4060025   0.200   0.8437  \n",
              "am           3.2727347  2.4911753   1.314   0.2064  \n",
              "gear         0.3593642  1.7813105   0.202   0.8425  \n",
              "carb        -0.1588580  0.9187435  -0.173   0.8648  \n",
              "---\n",
              "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
              "\n",
              "Residual standard error: 2.859 on 17 degrees of freedom\n",
              "Multiple R-squared:  0.8678,\tAdjusted R-squared:   0.79 \n",
              "F-statistic: 11.16 on 10 and 17 DF,  p-value: 1.227e-05\n"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        },
        {
          "data": {
            "text/html": [
              "<style>\n",
              ".dl-inline {width: auto; margin:0; padding: 0}\n",
              ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
              ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
              ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
              "</style><dl class=dl-inline><dt>RMSE</dt><dd>1.67422696953526</dd><dt>Rsquared</dt><dd>0.94365346180333</dd><dt>MAE</dt><dd>1.26093984902831</dd></dl>\n"
            ],
            "text/latex": [
              "\\begin{description*}\n",
              "\\item[RMSE] 1.67422696953526\n",
              "\\item[Rsquared] 0.94365346180333\n",
              "\\item[MAE] 1.26093984902831\n",
              "\\end{description*}\n"
            ],
            "text/markdown": [
              "RMSE\n",
              ":   1.67422696953526Rsquared\n",
              ":   0.94365346180333MAE\n",
              ":   1.26093984902831\n",
              "\n"
            ],
            "text/plain": [
              "     RMSE  Rsquared       MAE \n",
              "1.6742270 0.9436535 1.2609398 "
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "# Load the necessary library\n",
        "library(caret)\n",
        "\n",
        "# Load the mtcars dataset\n",
        "data(mtcars)\n",
        "\n",
        "# Split the dataset into a training set and a test set\n",
        "set.seed(42)\n",
        "trainIndex <- createDataPartition(mtcars$mpg, p = .8, list = FALSE, times = 1)\n",
        "train <- mtcars[trainIndex,]\n",
        "test <- mtcars[-trainIndex,]\n",
        "\n",
        "# Create a linear regression model\n",
        "model <- lm(mpg ~ ., data = train)\n",
        "\n",
        "# Print the summary of the model\n",
        "summary(model)\n",
        "\n",
        "# Make predictions on the test set\n",
        "predictions <- predict(model, newdata = test)\n",
        "\n",
        "# Calculate and print the RMSE and R-squared\n",
        "postResample(pred = predictions, obs = test$mpg)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n",
        "\n",
        "This code first loads the mtcars dataset and splits it into a training set and a test set. Then it creates a linear regression model and trains it on the training data. It prints a summary of the model, which includes the coefficients of the model and various metrics such as R-squared, Adjusted R-squared, and Residual Standard Error. It makes predictions on the test data and calculates the Root Mean Squared Error (RMSE) and R-squared of the predictions."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Advantages of Logistic Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Linear Regression also has several advantages:\n",
        "\n",
        "1. **Simplicity**: Linear regression is a simple model that is easy to understand and interpret. It provides a clear relationship between the dependent and independent variables.\n",
        "\n",
        "2. **Efficiency**: Linear regression is computationally inexpensive to train, making it a good choice for problems with a large number of features and instances.\n",
        "\n",
        "3. **Interpretability**: The coefficients in a linear regression model are interpretable. They represent the change in the dependent variable for a one-unit change in an independent variable, assuming all other variables are held constant.\n",
        "\n",
        "4. **Predictive Power**: Despite its simplicity, linear regression can be surprisingly powerful. In many real-world situations, the relationship between variables is approximately linear, and linear regression can capture these relationships effectively.\n",
        "\n",
        "5. **Basis for Many Methods**: Linear regression forms the basis for many other machine learning methods. It's a good starting point for more complex modeling.\n",
        "\n",
        "6. **Quantifying the relationships**: Linear regression is able to quantify the relationships between the features and the target variable, which can be very useful in many applications.\n",
        "\n",
        "7. **It can handle overfitting using techniques like Ridge or Lasso regularization**: These techniques add a penalty term to the loss function, which helps to reduce overfitting by decreasing the complexity of the model.\n",
        "\n",
        "8. **It works well on linearly separable data**: If the data is linearly separable, linear regression can model the relationship effectively."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Disadvantages of Logistic Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "While Linear Regression is a useful tool, it does have several disadvantages:\n",
        "\n",
        "1. **Assumption of Linearity**: Linear regression assumes a linear relationship between the independent and dependent variables. If this assumption is violated, the model's predictive performance can be poor.\n",
        "\n",
        "2. **Sensitive to Outliers**: Linear regression is sensitive to outliers in the data. Outliers can have a large effect on the regression line and can lead to a poor fit to the data.\n",
        "\n",
        "3. **Multicollinearity**: Linear regression requires that the independent variables are not highly correlated with each other, a condition known as multicollinearity. If this condition is violated, it can make the model's estimates and predictions unreliable.\n",
        "\n",
        "4. **Homoscedasticity Assumption**: Linear regression assumes that the variance of the errors is constant across all levels of the independent variables. This is known as the assumption of homoscedasticity. If the variance of the errors changes across the levels of the independent variables, then the prediction intervals will be inaccurate.\n",
        "\n",
        "5. **Independence of Observations**: The observations (rows) in your dataset should be independent of each other. If there is correlation among observations, such as in time series data or data collected in clusters, then linear regression may not be the best model choice.\n",
        "\n",
        "6. **Doesn't Handle Non-linear Relationships Well**: If the relationship between the variables is not linear, linear regression will not be able to accurately capture the relationship. More flexible models like polynomial regression, decision trees or neural networks might be more appropriate in these cases.\n",
        "\n",
        "7. **Overfitting**: Linear regression models can be prone to overfitting, especially if the model is too complex (i.e., if it has too many features or if the features are highly correlated). Regularization techniques can be used to prevent overfitting."
      ]
    },
    {
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      "metadata": {},
      "source": [
        "# Application of Linear Regression?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Linear Regression is a versatile algorithm that can be used in a variety of applications:\n",
        "\n",
        "1. **Real Estate Pricing**: Linear regression can be used to predict the price of houses based on features like the number of rooms, location, size, etc.\n",
        "\n",
        "2. **Stock Price Prediction**: Linear regression can be used in predicting the price of stocks based on historical stock price data.\n",
        "\n",
        "3. **Sales Forecasting**: Businesses can use linear regression to predict future sales based on trends and seasonality in historical sales data.\n",
        "\n",
        "4. **Risk Assessment in Insurance**: Insurance companies can use linear regression to predict the claims an insured person might make based on factors like age, medical history, etc.\n",
        "\n",
        "5. **Healthcare**: In healthcare, linear regression can be used to predict patient outcomes based on various health indicators.\n",
        "\n",
        "6. **Supply Chain**: Linear regression can be used to forecast demand and manage inventory in supply chain and logistics.\n",
        "\n",
        "7. **Economics**: Linear regression is used in economics to understand and quantify the relationships between different economic variables.\n",
        "\n",
        "8. **Marketing and Advertising**: Companies can use linear regression to understand the impact of advertising spend on sales, or to predict the success of a marketing campaign.\n",
        "\n",
        "9. **Academics and Research**: Linear regression is widely used in various fields of research to understand the relationship between different variables.\n",
        "\n",
        "10. **Credit Scoring**: Linear regression can be used to predict the credit score of a customer based on their financial history and other related factors."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "G1qobmLox7U0"
      },
      "source": [
        "# Reference:\n",
        "1. [Analytics Vidya]( https://www.analyticsvidhya.com/blog/2021/10/everything-you-need-to-know-about-linear-regression) "
      ]
    }
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